Related papers: Longest common subsequences in sets of words

We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We…

Fix two words over the binary alphabet $\{0,1\}$, and generate iid Bernoulli$(p)$ bits until one of the words occurs in sequence. This setup, commonly known as Penney's ante, was popularized by Conway, who found (in unpublished work) a…

We establish new upper bounds for the length of runs of consecutive positive integers each with exactly $k$ divisors, where $k$ is a given positive integer of some special forms. Also we have found exact values of the maximum possible runs…

Let $f (\cdot)$ be the absolute frequency of words and $r$ be the rank of words in decreasing order of frequency, then the following function can fit the rank-frequency relation \[ f (r;s,t) = \left(\frac{r_{\tt max}}{r}\right)^{1-s}…

Although 10^230 terms of Recaman's sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) {A(n), n >= 3} is defined as follows. Start with n, and…

Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with…

Let $A$ and $B$ be sets of words of length $n$ over some finite alphabet. Suppose that no suffix of a word in $A$ coincides with a prefix of a word in $B$. Then we show that the product of densities of $A$ and $B$ is upper bounded by…

The longest common subsequence (LCS) is a fundamental problem in string processing which has numerous algorithmic studies, extensions, and applications. A sequence $u_1, \ldots, u_f$ of $f$ strings s said to be an ($f$-)segmentation of a…

Let $(m, n, k)$ be a tuple of integers with the property that if $i \leq k$, then $m + i$ and $n + i$ have the same radical. Using a result on the abc Conjecture, we bound $k$ from above, improving a result of Balasubramanian, Shorey, and…

We prove a density version of the Carlson--Simpson Theorem. Specifically we show the following. For every integer $k\geq 2$ and every set $A$ of words over $k$ satisfying \[\limsup_{n\to\infty} \frac{|A\cap [k]^n|}{k^n}>0\] there exist a…

Calculating the length of a longest common subsequence (LCS) of two strings $A$ and $B$ of length $n$ and $m$ is a classic research topic, with many worst-case oriented results known. We present two algorithms for LCS length calculation…

A word $w_1w_2\cdots w_n$ is said to be up-down if $w_1 < w_2 >w_3 \cdots$. Carlitz and Scoville found the generating function for the number of up-down words over an alphabet of size $k$. Using properties of the Chebyshev polynomials we…

We consider the problem of finding, given two documents of total length $n$, a longest string occurring as a substring of both documents. This problem, known as the Longest Common Substring (LCS) problem, has a classic $O(n)$-time solution…

Given a finite word $w$ over a finite alphabet $V$, consider the graph with vertex set $V$ and with an edge between two elements of $V$ if and only if the two elements alternate in the word $w$. Such a graph is said to be word-representable…

We study the properties of the sequence of words $(B_i)$, where $B_1 = 101$ and $B_{i+1} = B_i C_i$ for $i \geq 1$, where $C_i$ is $B_i$ with the first $i$ symbols removed, and the infinite binary sequence ${\bf b} = 10101101011011101…

Let $ k \geq 2 $ be an integer. The $ k- $generalized Fibonacci sequence is a sequence defined by the recurrence relation $ F_{n}^{(k)}=F_{n-1}^{(k)} + \cdots + F_{n-k}^{(k)}$ for all $ n \geq 2$ with the initial values $ F_{i}^{(k)}=0 $…

We establish several recurrence relations and an explicit formula for V(n), the number of factorizations of the length-n prefix of the Fibonacci word into a (not necessarily strictly) decreasing sequence of standard Fibonacci words. In…

For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$,…

Two words have a reverse if they have the same pair of distinct letters on the same pair of positions, but in reversed order. A set of words no two of which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size of a…

Let $W^{(n)}$ be the $n$-letter word obtained by repeating a fixed word $W$, and let $R_n$ be a random $n$-letter word over the same alphabet. We show several results about the length of the longest common subsequence (LCS) between…